Plotting the graph of the quadratic function and examining the roles of the parameters $a, b, c$ in the function of the form $y = ax^2 + bx + c$

Plotting the graph of the quadratic function and examining the roles of the parameters $a, b, c$ in the function of the form $y = ax^2 + bx + c$

$a$ – the coefficient of $X^2$.

$b$ – the coefficient of $X$.

$c$ – the constant term.

- Let's examine the parameter $a$ and ask: Is the function upward or downward facing?
- Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
- Let's find the points of intersection with the $X$ axis by substituting ($Y=0$).
- Let's draw a coordinate system and first mark the vertex of the parabola.

Then, let's examine if the function is smiling or crying and mark the points of intersection with the $X$ axis that we found. Draw accordingly.

\( y=x^2 \)

Let's present the quadratic function and understand the meaning of each parameter in it.

$y=ax^2+bx+c$

$a$ – the coefficient of $X^2$

- -determines if the parabola will be a maximum or minimum parabola (sad or smiling) - determines the steepness – its width.

$a$ must be different from 0.

If $a$ is positive – the parabola is a minimum parabola – smiling

If $a$ is negative – the parabola is a maximum parabola – sad

The larger $a$ is – the narrower the function will be and vice versa.

$b$ - the coefficient of $X$

can be any number

indicates the position of the parabola along with $C$.

determines the slope of the function at the intersection point with the $Y$ axis

(not relevant to the material)

$c$ – the constant term

is not dependent on $X$

can be any number

indicates the position of the parabola along with $X$

determines the y-intercept

responsible for the vertical shift of the function

Wonderful. Now, let's move on to plotting the quadratic function.

- Let's examine the parameter $a$ and ask: is the function upward or downward facing?
- Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
- Let's find the points of intersection with the $X$ axis by substituting ($Y=0$).
- Let's draw a coordinate system and first mark the vertex of the parabola.

Then, let's examine if the function is smiling or sad and mark the points of intersection with the $X$ axis that we found. Draw accordingly.

**Let's see an example -**

In the following function:

$Y=-4X^2+4x+3$

- $a=-4$, negative. Therefore, the function is downward facing.
- Let's find the vertex of the parabola:

$X=\frac{-4}{-4*2}=\frac{-4}{-8}=\frac{1}{2}$

$y=-4*\frac{1^2}{2}+4*\frac{1}{2}+3$

$y=4$

Vertex of the parabola $(\frac{1}{2}, 4)$

- Let's find the intersection points with the $X$ axis

Substitute

$Y=0$

and we get:

$-4X^2+4x+3=0$

$X=-\frac{1}{2},1\frac{1}{2}$

- Draw a coordinate system, mark relevant points, and draw logically:

Test your knowledge

Question 1

\( y=x^2+10x \)

Question 2

\( y=x^2-6x+4 \)

Question 3

\( y=2x^2-5x+6 \)

$y=x^2+10x$

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

$y = ax²+bx+c$

Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.

Firstly, it seems that in this formula we do not have the C,

Therefore, we understand it is equal to 0.

$c = 0$

a is the coefficient of X², here it does not have a coefficient, therefore

$a = 1$

$b= 10$

is the number that comes before the X that is not squared.

$a=1,b=10,c=0$

$y=2x^2-5x+6$

In fact, a quadratic equation is composed as follows:

y = ax²-bx-c

That is,

a is the coefficient of x², in this case 2.

b is the coefficient of x, in this case 5.

And c is the number without a variable at the end, in this case 6.

$a=2,b=-5,c=6$

What is the value of the coefficient $b$ in the equation below?

$3x^2+8x-5$

The quadratic equation of the given problem is already arranged (that is, all the terms are found on one side and the 0 on the other side), thus we approach the given problem as follows;

**In the problem, the question was asked: **what is the value of the coefficient$b$in the equation?

**Let's remember** the definitions of coefficients when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$__are :__

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

**That is **the coefficient$b$is the coefficient of the **term in the first power** -$x$**We then examine **the equation of the given problem:

$3x^2+8x-5 =0$That is, the number that multiplies

$x$ is

$8$Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number$8$,

__Thus the correct answer is option d.__

8

What is the value of the coefficient $c$ in the equation below?

$3x^2+5x$

The quadratic equation of the given problem has already been arranged (that is, all the terms are on one side and 0 is on the other side) thus we can approach the question as follows:

**In the problem, the question was asked: **what is the value of the coefficient$c$in the equation?

**Let's remember** the definition of a coefficient when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$__are:__

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

**That is **the coefficient

$c$is the free term - and as such the coefficient of the **term is raised to the power of zero** -$x^0$(Any number other than zero raised to the power of zero equals 1:

$x^0=1$)

**Next we examine **the equation of the given problem:

$3x^2+5x=0$Note that there is no free term in the equation, that is, the numerical value of the free term is 0, in fact the equation can be written as follows:

$3x^2+5x+0=0$and therefore the value of the coefficient$c$ is 0.

__Hence the correct answer is option c.__

0

$y=x^2$

$a=1,b=0,c=0$

Related Subjects

- Quadratice Equations and Systems of Quadraric Equations
- Quadratic Equations System - Algebraic and Graphical Solution
- Solution of a system of equations - one of them is linear and the other quadratic
- Intersection between two parabolas
- Word Problems
- Properties of the roots of quadratic equations - Vieta's formulas
- Ways to represent a quadratic function
- Various Forms of the Quadratic Function
- Standard Form of the Quadratic Function
- Vertex form of the quadratic equation
- Factored form of the quadratic function
- The quadratic function
- Quadratic Inequality
- Parabola
- Symmetry in a parabola
- Finding the Zeros of a Parabola
- Methods for Solving a Quadratic Function
- Completing the square in a quadratic equation
- Squared Trinomial
- The quadratic equation
- Families of Parabolas
- The functions y=x²
- Family of Parabolas y=x²+c: Vertical Shift
- Family of Parabolas y=(x-p)²
- Family of Parabolas y=(x-p)²+k (combination of horizontal and vertical shifts)